Geostatistical investigations for suitable mapping of the water table: the Bordeaux case (France)

Abstract

Methodologies have been developed to establish realistic water-table maps using geostatistical methods: ordinary kriging (OK), cokriging (CoK), collocated cokriging (CoCoK), and kriging with external drift (KED). In fact, in a hilly terrain, when piezometric data are sparsely distributed over large areas, the water-table maps obtained by these methods provide exact water levels at monitoring wells but fail to represent the groundwater flow system, manifested through an interpolated water table above the topography. A methodology is developed in order to rebuild water-table maps for urban areas at the city scale. The interpolation methodology is presented and applied in a case study where water levels are monitored at a set of 47 points for a part urban domain covering 25.6 km² close to Bordeaux city, France. To select the best method, a geographic information system was used to visualize surfaces reconstructed with each method. A cross-validation was carried out to evaluate the predictive performances of each kriging method. KED proves to be the most accurate and yields a better description of the local fluctuations induced by the topography (natural occurrence of ridges and valleys).

Résumé

Des méthodologies ont été développées pour établir des cartes piézométriques réalistes en utilisant des méthodes géostatistiques : krigeage ordinaire (OK), cokrigeage (CoK), cokrigeage collocalisé (CoCoK) et krigeage avec dérive externe (KED). En fait, sur un terrain avec relief, lorsque les données piézométriques sont clairsemées sur de grandes surfaces, les piézométries obtenues par ces méthodes respectent les valeurs exactes aux piézomètres mais ne parviennent pas à représenter le système des écoulements souterrains, avec des interpolations de la nappe qui passent au-dessus de la surface topographique. Une méthode est développée de manière à bâtir les cartes piézométriques pour les zones urbaines, à l’échelle de la ville. La méthodologie d’interpolation est présentée et appliquée à un cas d’étude où les niveaux piézométriques sont suivis sur 47 points dans une zone urbaine de 26.5 km², à proximité de Bordeaux, en France. Pour sélectionner la meilleure méthode, un système d’information géographique a été utilisé afin de visualiser les surfaces construites avec chaque méthode. Une validation croisée a été réalisée pour évaluer les performances prédictives de chaque méthode de krigeage. La méthode KED est apparue la plus précise et apporte une meilleure description des fluctuations locales induites par la topographie (occurrence naturelle de crêtes et vallées).

Resumen

Se han desarrollado metodologías para establecer mapeos realistas del nivel freático utilizando métodos geoestadísticos: kriging ordinario (OK), cokriging (CoK), cokriging superpuestos (CoCoK) y kriging con deriva externa (KED). De hecho, en un terreno de colinas, cuando los datos piezométricos están esparcidamente distribuidos sobre grandes áreas, los mapas obtenidos del nivel freático por estos métodos proporcionan niveles de agua exactos en los pozos de monitoreo, fallan al representar el sistema de flujo de agua subterránea, que se manifiesta a través de un nivel freático interpolado por encima de la topografía. Se desarrolló una metodología para reconstruir mapas del nivel freático para las zonas urbanas a escala de la ciudad. Se presenta y aplica la metodología de interpolación en un caso de estudio en el que los niveles de agua son monitoreados en un conjunto de 47 puntos para una parte del dominio urbano cubriendo 25.6 km² cerca de la ciudad de Burdeos, Francia. Para seleccionar el mejor método, se utilizó un sistema de información geográfica para visualizar superficies reconstruidas con cada método. Se llevó a cabo una validación cruzada para evaluar la capacidad predictiva de cada método kriging. KED demuestra ser el más exacta y brinda una mejor descripción de las fluctuaciones locales inducidas por la topografía (ocurrencia natural de cordonoes y valles).

摘要

发展了方法论并采用地质统计方法建立了逼真的水位图:普通的克里格法,协同克里格法,并列协同克里格法及具有外部漂移的克里格法。事实上,在丘陵地带,当测压资料在很大的区域内分布很少时,用这些资料获取的水位图提供了监测井确切的水位,但代表不了地下水水流系统,这已经通过地形之上的内插水位所证明.发展了方法论就是要重建城区城市尺度的水位图.在法国波尔多市附近面积为25.6 km²的一片城区具有47个监测点位的研究案例中展示和应用了插值方法。为了选择最好的方法,采用地理信息系统对每个方法建立的水面形象化.进行了交叉确认,以评估每个克里格法的预测性能.具有外部漂移的克里格法证明最精确,能够更好地描述地形(天然出现的山脊和河谷)引起的局部波动。

Resumo

Metodologias têm sido desenvolvidas para definir mapas realistas do nível do lençol freático usando métodos geoestísticos: krigagem ordinária (KO), cokrigagem (CoK), cokrigagem coalocalizada (CoCoK), e krigagem com deriva externa (KDE). De fato, em um terreno acidentado, onde dados piezométricos são insuficientemente distribuídos através de grandes áreas, os mapas de nível do lençol freático obtidos por estes métodos proporcionam níveis exatos da água em poços de monitoramento, mas fracassam em representar o sistema de fluxo das águas subterrâneas, manifestado por um lençol freático interpolado sobre o terreno. A metodologia foi desenvolvida para reconstruir mapas de nível do lençol freático para áreas urbanas em escala municipal. A metodologia de interpolação foi apresentada e aplicada em um estudo de caso onde o nível da água foi monitorado no conjunto de 47 pontos para uma parte do domínio urbano, cobrindo 25.6 km² próximo da cidade de Bordéus, França. Para selecionar o melhor método, um sistema de informação geográfica foi usado para visualizar superfícies reconstruídas com cada método. Uma validação cruzada foi realizada para avaliar a performance preditiva de cada método de krigagem. KDE provou ser o mais preciso e produziu a melhor descrição da flutuação local induzida pela topografia (ocorrência natural de cordilheiras e vales).

Appendices

Appendix 1: Nomenclature

CoCoK:

collocated cokriging

CoCoKip:

collocated cokriging with imposed point

CoK:

cokriging

CoKip:

cokriging with imposed point

DEM:

digital elevation model

KED:

kriging with external drift

KEDip:

kriging with external drift with imposed point

MAE:

mean absolute error

ME:

mean error

OK:

ordinary kriging

OKip:

ordinary kriging with imposed point

QF:

Quaternary formations

MF:

Miocene formations

SK:

simple kriging

UK:

universal kriging

VSE:

variance of standardized error

Appendix 2

The mathematical expressions of the variogram model involve:

• 1. A coefficient which gives the order of magnitude of the variability along the vertical axis (homogenous to the variance). In the case of bounded functions (covariances), this value is simply the level of the plateau reached and is called the sill. The same concept has been kept even for the non-bounded functions and is called sill for convenience. The sill is equal to “C” in the following models.

• 2. When the function is bounded, it reaches a constant level (sill) after a given distance: this distance value is the range (or correlation distance in statistical language).

• 3. A third parameter required by some particular basic structures.

  Five variogram models, in which h is the lag distance, were examined in this report.

Power model

The power variogram is given by:

$$ \gamma (h)=C\times {\left(\frac{h}{a}\right)}^{\alpha } $$

(17)

where,

C :

is the multiplicative coefficient (also called sill for convenience)

a :

is the scale factor (also called range for convenience)

α :

is the exponent (shape parameter) defined as the third argument which must lie within [0,2]

Gaussian model

The Gaussian variogram is given by:

$$ \gamma (h)=C\times \left\{1- \exp \left[-{\left(\frac{h}{a/s}\right)}^2\right]\right\} $$

(18)

where,

C :

is the sill

a :

is the (practical) range

$$ s = 1.730818 $$

Cubic model

The Cubic variogram is given by:

$$ \gamma (h)=C\times \left[7\times {\left(\frac{h}{a}\right)}^2-\frac{35}{4}\times {\left(\frac{h}{a}\right)}^3+\frac{7}{2}\times {\left(\frac{h}{a}\right)}^5-\frac{3}{4}\times {\left(\frac{h}{a}\right)}^7\right] $$

(19)

where,

C :

is the sill

a :

is the range

Cauchy model

The Cauchy variogram is given by:

$$ \gamma (h)=C\times \left\{1-\frac{1}{{\left[1+{\left(\frac{h}{a}\right)}^2\right]}^{\alpha }}\right\} $$

(20)

where,

C :

is the sill

a :

is the range

α :

is the (positive) exponent defined as the third parameter

Stable model

The Stable variogram is given by:

$$ \gamma (h)=C\times \left\{1- \exp \left[-3{\left(\frac{h}{a}\right)}^{\alpha}\right]\right\} $$

(21)

where,

C :

is the sill

a :

is the range

α :

is the exponent defined as the third argument which must lie within [0;2]

Appendix 3

Ordinary kriging (OK)

OK estimates the expected value Z ^*_OK (u ₀) at location u ₀ as the weighted sum of the known data Z(u _α ) such that:

$$ {Z}_{\mathrm{OK}}^{*}\left({u}_0\right)={\displaystyle \sum_{\alpha =1}^n{W}_{\alpha }Z\left({u}_{\alpha}\right)} $$

(22)

where W _α are the weights chosen to minimize the prediction error variance by solving the following set of equations:

$$ \left\{\begin{array}{l}\begin{array}{lll}{\displaystyle \sum_{\alpha =1}^n{W}_{\alpha}\gamma \left({u}_{\alpha }-{u}_{\beta}\right)+\mu =\gamma \left({u}_{\beta }-{u}_0\right)}\hfill & for\hfill & \beta =1,\dots, n\hfill \end{array}\hfill \\ {}{\displaystyle \sum_{\alpha =1}^n{W}_{\alpha }=1}\hfill \end{array}\right. $$

(23)

here, γ(u _α  − u _β ) is the variogram between the data points u _α and u _β , γ(u _β  − u ₀) is the variogram between the data point β and the target point u ₀, and μ is a Lagrange multiplier introduced for the minimization of the error variance.

Cokriging (CoK) and collocated cokriging (CoCoK)

The cokriging is a natural extension of kriging in the case of multiple variables. In the case of two variables, the cokriging estimator equation can be written as

$$ {Z}_{\mathrm{CoK}}^{*}\left({u}_0\right)={\displaystyle \sum_{\alpha =1}^n{W}_{\mathrm{z}}^{\alpha }Z\left({u}_{\alpha}\right)}+{\displaystyle \sum_{\alpha =1}^m{W}_{\mathrm{s}}^{\alpha }S\left({u}_{\alpha}\right)} $$

(24)

in which n and m are the number of sampling points for primary and secondary variables respectively, W ^α_z and W ^α_s are the weights to be calculated solving the following equations:

$$ \left\{\begin{array}{l}\begin{array}{ll}{\displaystyle \sum_{\beta =1}^n{W}_{\mathrm{z}}^{\beta }{\gamma}_{\mathrm{z}\mathrm{z}}\left({u}_{\alpha }-{u}_{\beta}\right)}+{\displaystyle \sum_{\beta =1}^m{W}_{\mathrm{s}}^{\beta }{\gamma}_{\mathrm{z}\mathrm{s}}\left({u}_{\alpha }-{u}_{\beta}\right)}+{\mu}_1={\gamma}_{\mathrm{z}\mathrm{z}}\left({u}_{\alpha }-{u}_0\right)\hfill & \alpha =1,\dots n\hfill \end{array}\hfill \\ {}\begin{array}{ll}{\displaystyle \sum_{\beta =1}^n{W}_{\mathrm{z}}^{\beta }{\gamma}_{\mathrm{z}\mathrm{s}}\left({u}_{\alpha }-{u}_{\beta}\right)}+{\displaystyle \sum_{\beta =1}^m{W}_{\mathrm{s}}^{\beta }{\gamma}_{\mathrm{s}\mathrm{s}}\left({u}_{\alpha }-{u}_{\beta}\right)}+{\mu}_2={\gamma}_{\mathrm{z}\mathrm{s}}\left({u}_{\alpha }-{u}_0\right)\hfill & \alpha =1,\dots m\hfill \end{array}\hfill \\ {}{\displaystyle \sum_{\alpha =1}^n{W}_{\mathrm{z}}^{\alpha }}=1\hfill \\ {}{\displaystyle \sum_{\alpha =1}^m{W}_{\mathrm{s}}^{\alpha }}=0\hfill \end{array}\right. $$

(25)

where μ ₁ and μ ₂ are the Lagrange parameters which ensure unbiased conditions.

When the secondary variable is known at all nodes of a given estimation grid and at the data location of the primary variable, collocated cokriging can be used. In its strict sense, collocated cokriging makes use of the secondary variable only at the current point where the primary variable is to be estimated. In the multi-collocated form, it also makes use of the secondary variable at all points where the primary variable is available. The reader interested by the different approaches to collocated cokriging can refer to Rivoirard (2001). In this report, multi-collocated cokriging is used because it gives the more reliable and stable results. The collocated ordinary cokriging estimator is given by:

$$ {Z}_{\mathrm{CoCoK}}^{*}\left({u}_0\right)={W}_0S\left({u}_0\right)+{\displaystyle \sum_{\alpha =1}^n\left[{W}_{\mathrm{z}}^{\alpha }Z\left({u}_{\alpha}\right)+{W}_{\mathrm{s}}^{\alpha }S\left({u}_{\alpha}\right)\right]} $$

(26)

S(u ₀) is the secondary variable at the grid node u ₀ and S(u _α ) the secondary variable at the sample point of the primary variable.

Kriging with external drift (KED)

The KED involves performing standard kriging, based on the primary variable, considering that the external drift (overall shape of the regionalized phenomenon) is locally represented by the secondary variable. The KED estimator can be decomposed as a two-stage process: a generalized regression of the primary variable with the secondary variables followed by a simple kriging of the residuals (Rivest et al. 2008). The reader can refer to Desbarats et al. (2002); Wackernagel (2002); Bourennane and King (2003) for more details about KED. The basic hypothesis is that the expectation of the primary variable can be written as:

$$ E\left[Z(u)\right]={a}_0+{b}_1S(u) $$

(27)

where a ₀ and b ₁ are unknown. The additional universality conditions relating to external drift variables measured exhaustively in the field are integrated in the kriging system:

$$ \left\{\begin{array}{l}\begin{array}{lll}{\displaystyle \sum_{\beta =1}^n{W}_{\beta }K\left({u}_{\alpha }-{u}_{\beta}\right)-{\mu}_1-{\mu}_2S\left({u}_{\alpha}\right)=K\left({u}_{\alpha }-{u}_0\right)}\hfill & for\hfill & \alpha =1,\dots n\hfill \end{array}\hfill \\ {}{\displaystyle \sum_{\beta =1}^n{W}_{\beta }S\left({u}_{\beta}\right)=S\left({u}_0\right)}\hfill \\ {}{\displaystyle \sum_{\beta =1}^n{W}_{\beta }S\left({u}_{\beta}\right)=S\left({u}_0\right)}\hfill \end{array}\right. $$

(28)

where K(h) is the generalized covariance.